I believe that it is misleading to try to visualize higher-rank tensors. Think of vector as collections of 3 or 4 numbers, not as arrows. Then the algebraic generalization to matrices, rank-3 tensors, etc, is trivial.

I know what a scalar is.
I know what a vector is.
I know what a linear transformation is.

But what in the name of sweet aunt petunia is a rank 3 tensor?

Love,
Plx Mny

It is often recommended that you don't try to visualize a 3rd rank tensor, or any tensor for that matter, but to think of it merely as a multi-linear function which maps vectors and 1-forms to to the set of real numbers. The higher the rank the tougher the visualization becomes. In fact I never try to visualize a tensor myself.

There are always people who anyone never read. So what does David Hume have to do with this and why didn't you explain it when you posted his name? That's an irritating habit for you to fall into. :yuck:

I was wondering this same thing. I can't even visualize linear transformations. I keep wondering: does not being able to visualize it make it impossible to do things like GR if you're a "geometric thinker" like me as opposed to a "formula thinker" like... pretty much everyone else?

I'm really bad. My eyes spin in circles whenever I see a sum in sigma notation, and I have to write it out with the ellipsis before I understand what it's saying!

That holds true for any kind of transformation and not specifically to a linear one. A linear transformation is of the form

Y = aX + B

I.e. all linear transformations have this form.

Pete

That's not right. A linear transformation is not allowed to offset the vector (your B), and it can do a lot more than just scale the vector by a scalar "a". It can skew it, rotate it, and other things.

That's not right. A linear transformation is not allowed to offset the vector (your B), and it can do a lot more than just scale the vector by a scalar "a". It can skew it, rotate it, and other things.

That's interesting. That got me thinking and exploring Wikipedia. I guess the best way to visualize a general linear transformation is to visualize the three basis vectors of a 3D coordinate system and think of just changing/moving any/all of them in any way, with their tails remaining stuck together at the origin. Then I can picture the effect on any object living in the space "attached" to those vectors.

Then, a rank-3 tensor is like picturing those three arbitrarily changed/moved vectors in a different coordinate system, and applying a different linear transformation to each one.

The gradient of a rank 2 tensor (matrix) in a coordinate system (vector), is a rank 3 tensor. (GAMUDD) in example below.
Another example is the gradient of the metric in general relativity which is the connection.
An example of a 4th rank tensor is the Riemann curvature of spacetime. RUDDD in http://eigenmath.net/examples/bondi-metric.txt .